Event Detail

Fri Oct 20, 2006
60 Evans Hall, 4:10–6 PM
Logic Colloquium
Dana S. Scott (UC-Berkeley)
Duality in Projective Geometry

On four years (1989, 1993, 1997, 1998), I offered a one-semester course on algebraic curves in the classical complex projective plane. The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often effected by conic sections). What is not so well known is that this duality can be extended to higher-degree loci, relating point ranges (i.e., curves) to line configurations (i.e., envelopes of tangents).

In the plane over the complex numbers, a homogeneous polynomial f(x, y, z) of degree n, representing the point locus f(x, y, z) = 0, can be related to an envelope of degree m, say g(u, v, w) = 0, written in line coordinates, by regarding the symbols u, v, and w as the partial differential operators with respect to x, y, and z. In fact, the relationship is completely symmetric in that we can as well regard the symbols x, y, and z as the partial differential operators with respect to u, v, and w. Combining algebra with partial differentiation allows for symbolic proofs of properties of curves with simple conditions for representing singular points or tangents. These proofs can be materially aided with a judicious use of computer algebra.

Symbolic polynomial algebra really does give a middle way between synthetic and analytic geometry. The point is that symbolic computation allows one to name geometric objects by symbols and then to apply only those rules appropriate to the problem at hand. From the point of view of first-order logic, this approach can be described in terms of a decidable theory. We know that the field has a decidable theory, and hence the structure PQ = of points and lines of the complex projective plane under the incidence relation also has a decidable theory. But more generally the multi-sorted first-order theory of the structure PQ* == is decidable. Here Pn is the space of homogeneous xyz-polynomials of degree n over the complex field C, and Qn is the space of homogeneous uvw-polynomials of degree n. The binary operation f D g between xyz-polynomials and uvw-polynomials means that the polynomial of lower degree acts on the polynomial of higher degree by partial differentiation.